Why the Information Fractions Matter
A group sequential design allocates opportunities to examine accumulating evidence before the planned final analysis. The information fraction \(t_i\) locates look \(i\) on the information scale: \(t_i=0.40\), for example, means that the look is planned when 40% of the maximum information has accrued. Choosing the sequence of these fractions is a design decision. It determines when the study may stop early and which cumulative type I error targets the boundary calibration must meet.
A spending function specifies how much of the type I error budget is to have been spent by each information fraction. Thus, choosing the fractions and choosing the spending function are related but distinct tasks. The function describes the desired cumulative allocation over information time; the fractions specify the planned points at which that allocation is evaluated.
An increment is an allocation on the cumulative error scale, not generally the probability that the test statistic will cross the boundary at that look conditional on reaching it. The actual boundaries must be calibrated jointly so that the probability of crossing by each look matches its cumulative target.
Choosing and Checking a Fraction Sequence
A proposed sequence should be strictly increasing and should end at the final planned information fraction, conventionally \(t_K=1\). Each fraction should reflect a feasible analysis time and a useful decision point. Looks that are very close together offer little additional information between analyses but still require distinct boundaries. Widely separated looks leave more information to accrue between decisions, which can affect both the chance of early stopping and the boundary needed at the later look.
The information scale, rather than calendar time alone, is the relevant scale for a canonical group sequential design. A fixed calendar interval need not correspond to a fixed information increment. The earlier material on spending with unequal information increments establishes that unequal increments are permitted; the key design question here is where to place the planned fractions and what cumulative targets they imply.
Before calibration, check that the proposed fractions are compatible with the design assumptions: information is increasing, the final analysis is at the specified maximum information, and the canonical test-statistic model is appropriate. Under that model, the standardized statistics have mean zero and variance one under \(H_0\), with correlation between looks \(i\) and \(j\) given by \(\sqrt{t_i/t_j}\) for \(i<j\). This dependence matters: crossing events at successive looks are not independent.
Worked Example: Converting Fractions to Targets Under Linear Spending
Suppose a hypothetical two-sided design has \(\alpha=0.05\) and uses the linear spending function \(A(t)=\alpha t\). Its planned information fractions are \(t_1=0.20\), \(t_2=0.60\), and \(t_3=1.00\). Calculate the cumulative target and incremental target at each look.
At the first look, \(A(0.20)=0.05(0.20)=0.010\), so the increment is \(0.010-0=0.010\).
At the second look, \(A(0.60)=0.05(0.60)=0.030\), so the increment is \(0.030-0.010=0.020\).
At the final look, \(A(1.00)=0.05(1.00)=0.050\), so the increment is \(0.050-0.030=0.020\). The increments sum to \(0.010+0.020+0.020=0.050\), as required.
These numbers are cumulative targets and increments in the planned type I error spending. They are not, by themselves, critical values or marginal probabilities of crossing at each look.
How Spacing Changes the Boundary Calibration
For a two-sided design with symmetric efficacy boundaries, let \(c_i\) be the positive boundary at look \(i\). A crossing at that look occurs when \(|Z_i|\ge c_i\). Calibration aims to satisfy \(P(\text{a crossing by look }i\mid H_0)=A(t_i)\) at every planned look. At the first look there are no earlier crossing events, so its boundary has the direct form \(c_1=\Phi^{-1}(1-A(t_1)/2)\). At later looks, the boundary must account for the joint distribution of the current statistic and the event of not having crossed earlier.
For example, under the canonical model the two-look correlation is \(\sqrt{t_1/t_2}\). Changing \(t_1\) while leaving \(t_2\) fixed changes that correlation and changes the joint crossing probability. Consequently, the same cumulative target at the second look does not imply the same second-look boundary under two different look schedules.
The first-look calculation also shows the direct effect of an early target: a larger \(A(t_1)\) produces a smaller \(c_1\), making an early crossing easier. Later boundaries are not obtained by applying the same one-look formula to each incremental target. Their calibration must preserve the desired cumulative crossing probabilities across the sequence.
Worked Example: Comparing Early-Look Placement
Consider two hypothetical two-sided designs with \(\alpha=0.05\) and linear spending \(A(t)=0.05t\). Design A has fractions \(0.50,1.00\); Design B has fractions \(0.80,1.00\). Compare their first-look targets and boundaries.
For Design A, the first-look target is \(A(0.50)=0.05(0.50)=0.025\). Thus \(c_1=\Phi^{-1}(1-0.025/2)=\Phi^{-1}(0.9875)\approx2.241\).
For Design B, the first-look target is \(A(0.80)=0.05(0.80)=0.040\). Thus \(c_1=\Phi^{-1}(1-0.040/2)=\Phi^{-1}(0.9800)\approx2.054\).
The later first look in Design B has a larger cumulative target and a less stringent first boundary. Both designs still have the same final cumulative target, \(A(1)=0.050\). Their second-look boundaries need not be equal: the correlation with the first-look statistic differs, and the probability of having already crossed differs. The two first-look calculations are valid because each first-look statistic is standard normal under \(H_0\); the later boundaries require sequential calibration.
Evaluating a Spending Curve at Unequal Fractions
A curved spending function can allocate error very differently from a linear function. One commonly used O’Brien–Fleming-like two-sided spending curve is \(A(t)=2\{1-\Phi(z_{1-\alpha/2}/\sqrt{t})\}\), where \(z_{1-\alpha/2}=\Phi^{-1}(1-\alpha/2)\). It spends relatively little error early and approaches \(\alpha\) at \(t=1\). The design fractions still determine the evaluation points; unequal spacing does not change the definition of the curve.
Worked Example: Spending Between Two Unequally Spaced Looks
For a hypothetical two-sided design with \(\alpha=0.05\), use the O’Brien–Fleming-like spending curve above and consider adjacent looks at \(t=0.70\) and \(t=0.90\). The increment between them is the difference of the cumulative targets:
Thus, approximately 0.01968 of cumulative type I error is allocated between these information fractions, or about 1.968 percentage points. This increment is larger than the fraction gap \(0.90-0.70=0.20\) alone might suggest: the spending curve is nonlinear. To obtain the complete schedule, evaluate \(A(t_i)\) at every planned fraction and subtract adjacent cumulative targets. Boundary calibration then uses the cumulative targets, not just this interval increment.
A Short Justification for Using Cumulative Targets
The defining target at look \(i\) concerns the event of any crossing up to that look. Let \(C_i\) denote that event, and let \(C_0\) be the impossible event. The new crossing event at look \(i\) is \(C_i\setminus C_{i-1}\). Since this event is contained in \(C_i\), and \(C_i\) is the disjoint union of \(C_{i-1}\) and \(C_i\setminus C_{i-1}\),
If calibration sets \(P(C_i\mid H_0)=A(t_i)\) at each look, subtracting the previous target gives \(P(C_i\setminus C_{i-1}\mid H_0)=A(t_i)-A(t_{i-1})\). This explains why target increments add to the total cumulative spending. It does not make the increment a marginal tail probability such as \(P(|Z_i|\ge c_i\mid H_0)\): a path may already have crossed at an earlier look, and the statistics are dependent.
Diagnostics, Alternatives, and Limitations
Several checks can reveal a poorly specified schedule before or after calibration. Confirm that the fractions are strictly ordered, lie in \((0,1]\), and end at 1. Check that \(A(t_i)\) is nondecreasing and that the increments are nonnegative and sum to \(\alpha\), up to rounding. A negative increment usually indicates a nonmonotone spending function, an incorrectly ordered fraction sequence, or an arithmetic error. Also verify that the planned fractions are meaningful on the information scale; calendar dates alone do not establish their information values.
After calibration, verify that the resulting boundaries reproduce the intended cumulative targets under the assumed joint null distribution. If the computed crossing probabilities do not match \(A(t_i)\), the issue is not fixed by relabeling increments. Possible causes include using the wrong correlation structure, mixing one-sided and two-sided spending, or substituting marginal tail areas for cumulative crossing probabilities.
An alternative to spending-function designs is to specify a fixed set of critical boundaries directly. That can be appropriate when the analysis schedule and operating characteristics are already fixed, but it does not offer the same direct way to express cumulative error targets as a function of information. Conversely, a spending function does not make the choice of looks irrelevant: different schedules produce different evaluations of the function and different dependence among looks.
Information fractions must be planned or otherwise justified for the intended design. This tutorial does not infer them from sample sizes or event counts; those require their own information models. Nor does a correctly chosen sequence protect against misspecified test-statistic distributions, unplanned analyses, or changes to the design that are not reflected in calibration. Spending targets describe an error-control plan under stated assumptions, not a guarantee that every operational deviation will preserve it.
Common Mistake
Write the planned sequence in increasing order and confirm that the final fraction is 1.
Calculate \(A(t_i)\) at every planned fraction to obtain cumulative targets.
Subtract adjacent cumulative targets, then confirm that increments are nonnegative and sum to \(\alpha\).
Use the joint null distribution across looks to match each cumulative target; do not use later-look marginal tail areas as substitutes.
Check Your Understanding
Answer each question using cumulative targets, not isolated look probabilities.
- With \(\alpha=0.04\), linear spending, and fractions \(0.25,0.75,1.00\), what are the cumulative targets and increments?
- For a nondecreasing spending function, what does a negative increment suggest?
- Why can two designs with the same final fraction and final alpha have different second-look boundaries?
- At the first look, explain how a larger cumulative target changes the two-sided boundary.
Solutions. 1. Since \(A(t)=0.04t\), the cumulative targets are \(0.010,0.030,0.040\). The increments are \(0.010,0.020,0.010\), which sum to \(0.040\). 2. A negative increment conflicts with a nondecreasing spending function; check the fraction order, the function values, and the arithmetic. 3. The earlier look’s location changes the correlation between the statistics and the probability of an earlier crossing, so the later boundary required to meet the cumulative target can change. 4. The first-look boundary is \(c_1=\Phi^{-1}(1-A(t_1)/2)\). Increasing \(A(t_1)\) lowers the upper quantile and therefore lowers \(c_1\), making a crossing easier.